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Variations on the “exact factorization” theme

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Gonze, Xavier [UCL] Zhou, Jianqiang Sky [Laboratoire des Solides Irradiés, École Polytechnique, CNRS, CEA, Université Paris-Saclay, F-91128 Palaiseau, France] Reining, Lucia [Laboratoire des Solides Irradiés, École Polytechnique, CNRS, CEA, Université Paris-Saclay, F-91128 Palaiseau, France]

In a series of publications, Hardy Gross and co-workers have highlighted the interest of an “exact factorization” approach to the interacting electron-nuclei problem, be it time-independent or time-dependent. In this approach an effective potential governs the dynamics of the nuclei such that the resulting N-body nuclear density is in principle exact. This contrasts with the more usual adiabatic approach, where the effective potential leads to an approximate nuclear density. Inspired by discussions with Hardy, we explore the factorization idea for arbitrary many-body Hamiltonians, generalizing the electron-nuclei case, with a focus on the static case. While the exact equations do not lead to any practical advantage, they are illuminating, and may therefore constitute a suitable starting point for approximations. In particular, we find that unitary transformations that diagonalize the coupling term for one of the sub-systems make exact factorization appealing. The algorithms by which the equations for the separate subsystems can be solved in the time-independent case are also explored. We illustrate our discussions using the two-site Holstein model and the quantum Rabi model. Two factorization schemes are possible: one where the boson field feels a potential determined by the electrons, and the reverse exact factorization, where the electrons feel a potential determined by the bosons; both are explored in this work. A comparison with a self-energy approach is also presented.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès libre
Publication date 2018
Language Anglais
Journal information "The European Physical Journal B" - Vol. 91, no.10, p. - (2018)
Peer reviewed yes
Publisher Springer Nature America, Inc
issn 1434-6028
e-issn 1434-6036
Publication status Publié
Affiliation UCL - SST/IMCN/NAPS - Nanoscopic Physics
Keywords Electronic ; Optical and Magnetic Materials ; Condensed Matter Physics
Links
  1. A. Fetter, J. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 2000)
  2. Mahan Gerald D., Many-Particle Physics, ISBN:9781441933393, 10.1007/978-1-4757-5714-9
  3. R. Martin, L. Reining, D. Ceperley, Interacting Electrons: Theory and Computational Approaches (Cambrige University Press, Cambridge, 2016)
  4. M. Di Ventra, Non-equilibrium Green’s function formalism, in Electrical Transport in Nanoscale Systems (Cambrige University Press, 2008), p. 209
  5. Kotliar G., Savrasov S. Y., Haule K., Oudovenko V. S., Parcollet O., Marianetti C. A., Electronic structure calculations with dynamical mean-field theory, 10.1103/revmodphys.78.865
  6. Azumi Tohru, Matsuzaki Kazuo, WHAT DOES THE TERM "VIBRONIC COUPLING" MEAN?, 10.1111/j.1751-1097.1977.tb06918.x
  7. R. Martin, Electronic Structure: Basic Theory and Practical Methods (Cambridge University Press, Cambridge, U.K., 2008)
  8. M. Born, K. Huang, The Dynamical Theory of Crystal Lattices (Oxford University Press, London, 1954)
  9. Hunter Geoffrey, Conditional probability amplitudes in wave mechanics, 10.1002/qua.560090205
  10. N.I. Gidopoulos, E. Gross, https://doi.org/arXiv:cond-mat/0502433 [cond-mat.mtrl-sci] (2005)
  11. Gidopoulos N. I., Gross E. K. U., Electronic non-adiabatic states: towards a density functional theory beyond the Born-Oppenheimer approximation, 10.1098/rsta.2013.0059
  12. Abedi Ali, Maitra Neepa T., Gross E. K. U., Exact Factorization of the Time-Dependent Electron-Nuclear Wave Function, 10.1103/physrevlett.105.123002
  13. A. Abedi, N.T. Maitra, E.K.U. Gross, J. Chem. Phys. 137, 22A530 (2012)
  14. Berry M. V., Quantal Phase Factors Accompanying Adiabatic Changes, 10.1098/rspa.1984.0023
  15. Suzuki Yasumitsu, Abedi Ali, Maitra Neepa T., Yamashita Koichi, Gross E. K. U., Electronic Schrödinger equation with nonclassical nuclei, 10.1103/physreva.89.040501
  16. Cederbaum Lorenz S., The exact wavefunction of interacting N degrees of freedom as a product of N single-degree-of-freedom wavefunctions, 10.1016/j.chemphys.2015.05.021
  17. Hunter Geoffrey, The exact one-electron model of molecular structure, 10.1002/qua.560290209
  18. Schild Axel, Gross E. K. U., Exact Single-Electron Approach to the Dynamics of Molecules in Strong Laser Fields, 10.1103/physrevlett.118.163202
  19. Bohm David, A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I, 10.1103/physrev.85.166
  20. Abedi Ali, Agostini Federica, Gross E. K. U., Mixed quantum-classical dynamics from the exact decomposition of electron-nuclear motion, 10.1209/0295-5075/106/33001
  21. Agostini Federica, Abedi Ali, Gross E. K. U., Classical nuclear motion coupled to electronic non-adiabatic transitions, 10.1063/1.4902225
  22. Min Seung Kyu, Agostini Federica, Gross E. K. U., Coupled-Trajectory Quantum-Classical Approach to Electronic Decoherence in Nonadiabatic Processes, 10.1103/physrevlett.115.073001
  23. Agostini Federica, Min Seung Kyu, Abedi Ali, Gross E. K. U., Quantum-Classical Nonadiabatic Dynamics: Coupled- vs Independent-Trajectory Methods, 10.1021/acs.jctc.5b01180
  24. Min Seung Kyu, Agostini Federica, Tavernelli Ivano, Gross E. K. U., Ab Initio Nonadiabatic Dynamics with Coupled Trajectories: A Rigorous Approach to Quantum (De)Coherence, 10.1021/acs.jpclett.7b01249
  25. Requist Ryan, Gross E. K. U., Exact Factorization-Based Density Functional Theory of Electrons and Nuclei, 10.1103/physrevlett.117.193001
  26. Requist Ryan, Proetto César R., Gross E. K. U., Asymptotic analysis of the Berry curvature in the E⊗e Jahn-Teller model, 10.1103/physreva.96.062503
  27. Tully John C., Molecular dynamics with electronic transitions, 10.1063/1.459170
  28. Granucci Giovanni, Persico Maurizio, Critical appraisal of the fewest switches algorithm for surface hopping, 10.1063/1.2715585
  29. Ehrenfest P., Bemerkung �ber die angen�herte G�ltigkeit der klassischen Mechanik innerhalb der Quantenmechanik, 10.1007/bf01329203
  30. Khosravi Elham, Abedi Ali, Maitra Neepa T., Exact Potential Driving the Electron Dynamics in Enhanced Ionization ofH2+, 10.1103/physrevlett.115.263002
  31. Khosravi Elham, Abedi Ali, Rubio Angel, Maitra Neepa T., Electronic non-adiabatic dynamics in enhanced ionization of isotopologues of hydrogen molecular ions from the exact factorization perspective, 10.1039/c6cp08539c
  32. Suzuki Yasumitsu, Abedi Ali, Maitra Neepa T., Gross E. K. U., Laser-induced electron localization in H2+: mixed quantum-classical dynamics based on the exact time-dependent potential energy surface, 10.1039/c5cp03418c
  33. Fiedlschuster T., Handt J., Gross E. K. U., Schmidt R., Surface hopping in laser-driven molecular dynamics, 10.1103/physreva.95.063424
  34. Albareda Guillermo, Abedi Ali, Tavernelli Ivano, Rubio Angel, Universal steps in quantum dynamics with time-dependent potential-energy surfaces: Beyond the Born-Oppenheimer picture, 10.1103/physreva.94.062511
  35. Holstein T, Studies of polaron motion, 10.1016/0003-4916(59)90002-8
  36. Holstein T., Studies of polaron motion, 10.1016/0003-4916(59)90003-x
  37. Fulton Robert L., Gouterman Martin, Vibronic Coupling. I. Mathematical Treatment for Two Electronic States, 10.1063/1.1701181
  38. Lin Zijing, Han Rongsheng, Kelin Wang, Analytical Solutions of the Two-Site Holstein Model by the Coherent State Expansion, 10.1142/s0217979203022271
  39. McKemmish Laura K., McKenzie Ross H., Hush Noel S., Reimers Jeffrey R., Quantum entanglement between electronic and vibrational degrees of freedom in molecules, 10.1063/1.3671386
  40. Braak D., Integrability of the Rabi Model, 10.1103/physrevlett.107.100401
  41. Bersuker Isaac B., Pseudo-Jahn–Teller Effect—A Two-State Paradigm in Formation, Deformation, and Transformation of Molecular Systems and Solids, 10.1021/cr300279n
  42. Säkkinen Niko, Peng Yang, Appel Heiko, van Leeuwen Robert, Many-body Green’s function theory for electron-phonon interactions: Ground state properties of the Holstein dimer, 10.1063/1.4936142
  43. Vandaele Eva R J, Arvanitidis Athanasios, Ceulemans Arnout, The quantization of the Rabi Hamiltonian, 10.1088/1751-8121/aa5bc2
  44. H. van der Vorst, Iterative Krylov Methods for Large Linear Systems (Cambridge University Press, Cambridge, 2009)
  45. Hestenes M.R., Stiefel E., Methods of conjugate gradients for solving linear systems, 10.6028/jres.049.044
  46. W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes: The Art of Scientific Programming (Cambrige University Press, Cambridge, 1988)
  47. Teter Michael P., Payne Michael C., Allan Douglas C., Solution of Schrödinger’s equation for large systems, 10.1103/physrevb.40.12255
  48. Payne M. C., Teter M. P., Allan D. C., Arias T. A., Joannopoulos J. D., Iterative minimization techniques forab initiototal-energy calculations: molecular dynamics and conjugate gradients, 10.1103/revmodphys.64.1045
  49. I.G. Lang, Y.A. Firsov, Sov. Phys. JETP 16, 1301 (1963)
  50. Langreth David C., Singularities in the X-Ray Spectra of Metals, 10.1103/physrevb.1.471
  51. Agostini Federica, Abedi Ali, Suzuki Yasumitsu, Min Seung Kyu, Maitra Neepa T., Gross E. K. U., The exact forces on classical nuclei in non-adiabatic charge transfer, 10.1063/1.4908133
Bibliographic reference Gonze, Xavier ; Zhou, Jianqiang Sky ; Reining, Lucia. Variations on the “exact factorization” theme. In: The European Physical Journal B, Vol. 91, no.10, p. - (2018)
Permanent URL http://hdl.handle.net/2078.1/214079